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Journal of Pure and Applied Algebra 18 (1980) 149-164 0 North-Holland Publishing Company

GALOIS THEORY AND THEATERS OF ACTION IN A

John F. KENNISON* Clark University. Worcester, MA 01610, USA

Communicated by M. Barr Received 8 October 1978

0. Introduction

Constructing a for geometric fields in a topos presents certain difficulties. If K E F is a Galoisian extension in a topos 5Z,then one can construct the internal of K-automorphisms of F but it might be trivial even when K f F (see Example 6.6). In the topos of presheaves over a category C, a appears to be a presheaf over Cop. There are similar difficulties with splitting a polynomial over a field K in a topos. If we are splitting x2 + 1 over K in the topos of Sets, then we usually use a dichotomous procedure: if x2+ 1 is irreducible we construct K[i], if .r* + 1 is reducible we use K. This procedure is problematic for topoi since irreducibility is not a coherent condition (i.e., not geometric in the sense of [3]). A procedure for splitting polynomials is given in Section 3. See Example 6.3 also. Of course the field containing the added roots lives in a new topos, and this is an instance of the Cole Spectrum (see [l, 3,7]).

1. Theaters of action

A group in a reasonably complete topos 8 can be regarded as a left exact functor C + 8 where C is the dual of the category of finitely presented groups (see [2]. Note that “left exact” means finite limit preserving.) A Galois group should really be regarded as a profinite group by which we mean a left exact functor Go + 8 where Go is the category of finite groups. We could construct a Galois theory based on profinite groups but there would be complications concerning the definition of “closed .” [This is discussed in Section 5 and related to a curious open question of whether or under what conditions “co-equivalence relations” are “co-effective” for profinite groups.] It is more convenient to base our Galois theory on the concept of a

* The author is pleased to thank Sussex University for providing a stimulating place to work on this material during his recent Sabbatical leave. The author also thanks the National Science Foundation for support under NSF Grant MCS77-03482AOl.

149 150 J.F. Kennison rhea&r which incorporates the notion of a profinite group together with an object (or “theater of action”) on which the group acts. Thus a will be an example of a theater since (in Sets) the profinite Galois group acts on it. Observe that when we split x2 + 1 over the reals to get the complex field, there is no intrinsic way to distinguish i from -i. The Galois group serves to measure this “non-dis- tinguishability.” The concept of a theater is based on the idea of axiomatizing the positive notion of “distinguishability.” To illustrate the definition below, imagine that we have a Galois group acting on a field and define d(x, y) (x is “distinguished from” y) to mean that no member g of the Galois group sends x to y. Further define d,(,?, y’), where x’, y are n-tuples, to mean that there is no g for which g(f) = 7. Also define the “orbit control relation” O,,(x, g), for 6 an n-tuple, to be true when the orbit of x under the Galois group is contained in (61,. . . , b,).

Notation. 3 shall denote an n-tuple (for some n) of variables (xi, . . . , x,,). If u is in S, (the symmetric group on {1,2, . . . , n}), then Zmis the n-tuple (x,,,,, . . . , x,(,,). Given f and Q, then i * a shall be the (n + l)-tuple (xi, . . . ,x,, a). Similarly, if i is an n-tuple and y is a k-tuple, then x’ * 9 shall denote the obvious (n + k)-tuple. If f is a function (symbol), then f(Z) shall be the n-tuple (f(xl), . . . , f(x,)).

Definition. A theater in a topos $ is an object A together with (for each n) relations d, (of arity 2n) and 0, (of arity n + 1) such that the following conditions (called axioms) hold: Axiom A (pre-apartness). (Al) Not d,,(f, 2). (A2) d,(f, 1) iff d,(y, x‘). (A3) d,(f, 2) implies d,(K 1) or d,,(j, 2). (Comment. These axioms imply that for each n the relation ‘*Not d,” is an equivalence relation on A”. There is actually a countable infinitude of axioms given by the above axiom schemes). Axiom B. (Bl) (For each u in S,) d,(f, j? iff d,L Yc). (B2) d,(.F, 1) implies d,+l(.f * a, 7 * b). [So by induction we can also infer d,+k(i * d, Y * b).] (B3) d,(x, Y) iff d,+lLf * xn, Y * ~“1. Axiom C. If x1 =x2, then d2(xl, x2, yl, yz) or YI = YZ. (Comment. See the “Remark” following Proposition 1.4, below.) Axiom D. “Every x has an orbit.” That is:

V 5,. . . , b,, Onk 6). n

(This is an infinite disjunction; the axiom entails the assumption that the relevant sup exists.) Axiom E. O,(a, 6) and /jid,+*(f * a, y * bi) imply d,(f, >‘). Gal& theory and theaters of action in a topos 151

Axiom F. (Fl) O,,(X, &) implies O,,,(X, E) if the “set of 6’s is contained in the set of c’s.” This could be formalized by the analogues of Bl, B2, B3. (F2) Oncl(x, 6 * a) and dl(x, a) imply O”(x, 6).

Definition. Sometimes we omit the subscript n and write d(Z, 7) or 0(x, &), we further say “(d, 0) is theater structure on A,” etc. We define (A, d, 0) to be a prefheafer if the first five axioms (A-E) are satisfied. If (A, d, U) is a pre-theater then there is a unique way to define 0 so that (A, d, 0) is a theater. Uniqueness is proven below (1.3), the existence is obtained by 0(x, 6:, iff 3? and (T with V(x, (6 * ?),) and Ai [d(x, ci) or Vi C; = bi].

1.1 Proposition. In the topos of Sets, (A, d, 0) is a theater precisely when there is a profinite group G acting continuously on the (discrete) Set A such that d(T, jj) ifi g(2) f y for all g E G and O(a, i) iff for every g E G there exists an i with g(u) = bi. Moreover, if we assume that G acts effectively on A (meaning that g(a) = a for all a implies g is the group identity), then G and itsprofinite topology are determined to within .

Proof. If the profinite group G acts continuously on the set A, then it is clear that axioms A-F are satisfied when d, 0 are interpreted as indicated. Conversely, let (A, d, 0) be a theater in Sets. Define gc A XA to be pre- admissible if d(f, y) implies that (xi, y,)& g for at least one i. We say that g is admissible if it is maximally pre-admissible. Every pre-admissible relation is a function from a subset of A into A in view of axiom C. We claim that if g is admissible then the domain of g is all of A. Assume that a in A is not in the domain of g. Then for each b in A there must exist (2, p) with d(Y * a, y* 6) and g(Z) = 7, otherwise g u {(a, b)} is pre-admissible. Choose b so that O(a, 6), then one can readily find (2, y) with d(i * a, 7 * bj) for each j and g(Z) = p. By axiom E we obtain a contradic- tion, proving the claim. Observe that g is pre-admissible iff the inverse relation, g-’ is pre-admissible. The same is true of admissible relations so the above arguments show that if g is admissible then g is one-to-one and onto. The identity function is admissible by (Al) and a composition of admissible functions is admissible by (A3). It is easily shown that d (i, jj) iff g(Z) # J for all admissible g. (Since d (a, ~7)fails iff {(xi, yi)} is pre-admissible.) We also claim that O(a, &) iff the orbit of a is contained in PI,. . . , b,}. For if g(a) = c and if c # bi for all i, then d(a, a, c, bi) and hence d(a, c) by axioms C and E, contradicting g(a) = c. Conversely, if O(a, 6) we can then eliminate those bi for which d(a, bi), by (Fl) and (F2). Thus we can assume that O(a, &) where each bi is in the orbit. Hence if 5 contains the entire orbit we have O(a, E) by (Fl). In order for G to act continuously on A its topology must be at least as big as the topology of pointwise convergence with basic open sets W(d, 6) + {g/g(&) = b). Since 152 J.F. Kennison the orbits are finite the topology of pointwise convergence can be shown to be compact (and Hausdorff) so there is no strictly larger compact topology. Let G be the group of all admissible functions. It remains to show that no other group H can be compact, act continuously and effectively, and account for (d, 0) in the required way. If H acts effectively then H can be regarded as a group of permutations on A and, to account for d, we must have H E G. Since the basic open sets W(G, b) must intersect H in open sets (as H acts continuously) H maps continuously to G. Since H is compact, H is a closed subgroup of G. If W(ci, 6) is a non-empty open subset which misses H there would be a contradiction concerning d(a’, 6).

1.2. Corollary. Any geometric statement about the relations d, 0 which is true when interpreted about profinite groups acting on sets, is valid for a theater in any topos which has enough sups to analyze the statement.

Proof. A theater is defined as a model of a separable theory (see Makkai-Reyes [6, p. 1801). [There are countably many relations and axioms and the infinite dis- junctions are countable, too.] The corollary follows by 5.2.3 in [6, pp. 162-1631.

1.3. Corollary. If (d, 0) and (d’, 0’) are theater structures on A, then d = d’ implies O=O’.

Proof. Follows from 1.2 and the proof of 1.1. The admissible functions are defined only in terms of d.

1.4. Proposition. If (A, d, 0) is a theater, then A is decidable (i.e., the diagonal subobject A = {(a, a)} E A x A has a Boolean complement, I).

Proof. Let I = {(a, b)(d( a, a, a, b)}. The proof follows from 1.2, and also directly from (Al) and C.

Remark. Proposition 1.4 indicates that the given definition of “theater” is limited as decidability is generally a rather severe restriction. Geometric fields are always decidable (as either a = b or a -b is invertible) but for further applications axiom C should probably be modified to something like d(a, a, a, 6) or a is “apart from” 6.

Definition. A theater homomorphism f: (A, d, 0) --, (A’, d’, 0’) is a morphism f: A + A' such that d(K, 1) implies d’(f(.F), f(y)) and O(a, 6) implies O’(f(a), f(g)).

Theater homomorphisms are necessarily mono as they must preserve the relation I defined in 1.4. It is readily shown that f is an extremal mono (in the category of theaters in some fixed topos) precisely when d(Z, y’) iff d’( f (a), f(l)). Also f is an epi (for theater homomorphisms) precisely when f is an epi (hence an isomorphism) from A to A’ in the topos. [In any category f is an extremal mono iff f = gh and h epi imply h is an isomorphism.] Galois theory and theaters ofaction in a topos 153

Theater homomorphisms have essentially unique factorizations as me where m is extremal mono and e is epi. These assertions are readily proven in Sets and the proofs follow from the Makkai-Reyes results applied to the appropriate theories (whose models are diagrams of theaters and theater homomorphisms). We reserve the term subtheater for a subobject m: A + B where m is extremal mono (in the category of theaters of a fixed topos). Quotient theaters are defined as equivalence classes of epis in the usual way. The quotients of (A, d, 0) correspond to the theater structures (d’, 0’) on A for which d Ed’ (it follows that 0 E 0’). The following proposition is obvious.

1.5. Proposition. Let (A, d, 0) be a theater. The quotient theaters of (A, d, 0) correspond precisely to relations d’= {d;} f or which d C_d’ and (A, d’, 0) is a pre-theater. In the topos of Sets, assume that (A, d, 0) is the theater given by the effective action of the profinite group G on A. Then the quotients of (A, d, 0) correspond precisely with the theaters given by closed of G.

Proof. The first statement follows from the previous discussion. The second state- ment follows from the proof of 1.1.

Definition. If (A, d, 0) is a theater and if B is a subobject of A we define dB by

dB(Z, y’) iff (for some m) 3b,, . . . , 6, in B with d(.Z * 6; j * g).

(The phrase “for some m” could be replaced by the countable disjunction symbol, V,.) In Sets, if (A, d, 0) corresponds to the action of G on A, then ds is the quotient theater corresponding to the action of the closed subgroup H where g E H iff g\B is the identity. In general it is easy to verify that ds produces a quotient theater as described in 1.5. We define OB so that (A, de, OB) is a theater.

Definition. If (A, d, 0) is a theater, then the fixed point subobject of A is the solution subobject determined by the condition Oi(x, x). We denote this subobject by FixPt(d). It is easily shown that this has the obvious meaning in the case of Sets. Thus we can discuss actions of profinite groups (= theaters), closed subgroups (= quotient theaters), subgroups fixing a subobject (= dB) and fixed point subobjects in any topos and so are prepared for Galois theory. The following proposition is obvious (for example, it suffices to prove it for Sets).

1.6. Proposition. Let (A, d, 0) be a theater and B a subobject of A. Then

dFixPt(d)= d and B E FixPt(dB).

Moreover, for quotients d’ of d one has

B c FixPt(d’) iff ds cd’. 154 J.F. Kennison

2. Field extensions

Let K c F be geometric fields in a topos 8. (For each n) define K-Sep(d) as the coherent condition for an n-tuple B of F which states that:

(X-al)****-(X- a,) is a separable polynomial over K.

(This means that ai # ai for i # j and that the coefficients, or the appropriate symmetric functions of a r, . . . , a,, all lie in K.) We say that F is a Guloisiun extension of K iff for all x E F there exists, for some n, an n-tuple d with ur = x and K - Sep(b). This is a geometric condition given by:

v gal, * * * 1 an, K -Sep(d) and x = al. ”

It holds in the topos of Sets precisely when F is separably algebraic and normal over K. (It is geometric in the theory whose models are fields with a distinguished subfield.) If F is Galoisian extension of K we then define d,(n, y’) to mean that there is a polynomial P over K in n indeterminates for which P(Z) = 0 but P(y) # 0. [This is expressible geometrically by a countable disjunction.] We define O(u, &) to mean that there exists 8 with a L = a and each ai equal to some bi for which K - Sep(a’) holds. We call (d, 0) the Gulois theater of Fouer K and denote it by Gal.Tht.(F/K).

2.1. Lemma. Let F be a Guloisiun field extension of K in the topos of Sets. Then Gal.Tht.(F/K) is in fact a theater and corresponds to the action of the profinite Gulois group of F over K.

Proof. Everything is readily verified except for the assertion that if i and y are n-tuples of elements of F and one never has g(Z) = y for any g E G, the Galois group, then there exists a polynomial P(tl, . . . , t,) in n indeterminates over K with P(Z) = 0 and P(y) # 0. We prove this by induction on n. The case n = 1 follows by letting P be the minimal polynomial of x. Suppose the result is true for (n - l)-tuples. Then we may as well assume that we can fix some g E G with g(xi) = yi for i = 1, . . . , (n - 1). Let L be the subfield of F generated by K and x1,. . . , x,_~. Then there can be no L-automorphism h of F for which h(x,) = g-‘(y,) [for if so then gh E G and gh(i) = jj.1 By the case n = 1 there is a polynomial POin one indeterminate, t, over L with P,.,(x,,) = 0 and PO(g-‘(y,)) # 0. But the elements of L can be represented as polynomials in (n - 1) indeterminates over K, evaluated at (x1, . . . , x,-r). So clearly there is a polynomial P in n indeterminates such that PO(~) = P(xl, . . . , X,-I, 0. Now P(Z) = 0 and P(y) f 0 as g-‘(P(y)) = P(xl, . . . I Xn-1, g-‘(Y”)) = &(g-L(YnN f 0.

2.2. Theorem. Let K E F be a Guioisiun extension in a topos 8. Let (F, d, 0) = Gal.Tht.(F/K). Then (F, d, 0) is a theater and there is a one-to-one correspondence between intermediate subfields L and quotient theaters of (F, d, 0). That is: Galois theory and theaters of action in a topos 155

(2.2.1) If (F, d’, 0’) is a quotient of (F, d, 0) and if L = FixPt(F, d’, 0’), then L is an intermediate subfield and d’ = dr. (2.2.2) If L is an intermediate subfield, then L = FixPt(F, d‘, Ot) and (F, dL, OL) = Gal.Tht.(F/L).

Proof. (2.2.1) is a geometric statement in the theory of a Galoisian field extension together with a quotient theater of the Galois theater. (2.2.2) is a geometric statement in the theory of a Galoisian field extension together with an intermediate field. Both of these theories are separable in the sense of Makkai-Reyes [6, p. 1801. Both statements follow in Sets from Krull’s version of the fundamental theorem of Galois theory (for possibly infinite extensions using the profinite group and letting intermediate fields correspond with closed subgroups, which are quotient theaters by 1.5).

2.3. Proposition. Let F be a geometric field in a topos 8 and let (F, d, 0) be a theater. Assume that (d, 0) “preserves the ring operations” meaning that zero and one are fixed points and that:

dj(x, y, x + y, a, b, c) or c = a + b

and similarly for subtraction and multiplication. Let K = FixPt(F, d, 0). Then K is a subfield, F is Galoisian over K and (F, d, 0) = Gal.Tht.(F/K).

Proof. The result is easily verified in Sets, etc.

3. Splitting fields and root permutation

In this section we relate the Galois theater to “permutations of the roots.” To do this let K be a field in a topos 8 and consider a given manic polynomial

P(c)= tn+Cltn-l+* * *+c,_~t+c”

where t is an indeterminate and we assume that the coefficients cl, . . . , c,, are global elements of K. [Greater generality and more cluttered details can be obtained by first passing to the generic polynomial of degree n which does have global coefficients.] We wish to show how the Galois theater of any extension in which P splits is related to root permutations of P. To do this we shall construct the splitting field of P over K. This shall be a generic field extension which lives in a “spectrum” or topos over 8 as in Cole’s Theorem (see [3, Theorem 6.581, [l, 71). First we define the “splitting ring” by adjoining to K a set 7 = {r,, . . . , r”} of roots of P [that is, we require precisely those relations which are needed to make P = fl (t -ri), so cl = -1 ri, etc.] This generates a ring K(i). The symmetric group S, acts on K(7) in the obvious way and K(T) lives in ZYsn(the topos of S,, -objects of 8. K(F) in 8’” is the generic ring extension of K in which P splits. 156 J.F. Kennison

We proceed to make K(?) into a field. At this point we would normally require that P be irreducible, but this is a non-geometric requirement. Instead we assume that P is separable which means that there are polynomials M, N with MP + AP’ = 1 where P’ is the derivative of P. The advantage of this follows from:

3.1. Lemma. Let P be a separable polynomial. Then K(i) as defined above is a commutative (von Neumann) regular ring (both in 8 and in ZYSn). That is, K(7) satisfies the condition that for all x there exists y with xyx = x.

Proof. It suffices to prove this in Sets. We shall actually show that K(7) is a finite product of fields (in Sets). Note that K(P) is obtained by first adjoining any root rl of P to get K(r,) then adjoining (r2,. . . , r,) which satisfy the obvious conditions. It is readily shown that K(rl) is a product of fields, say K(r,) = n Fi. (Consider the irreducible components of P which must be distinct). Now there is a polynomial R with coefficients in K(rI) for which P(t) = (t - rl)R(t) (where t is indeterminate) as R can be constructed component-wise. Finally K(P) is obtained by splitting R over K(rl), hence over each Fj and the result follows by induction on n.

3.2. Proposition [Scott]. If R is a commutative regular ring in a topos %, then the quotient field of R can be constructed as follows: Let B be the Boolean algebra of idempotents of R. Then a B-sheaf is an object e: A + B, over B, together with a restriction operation a 16 for a E A, b E B for which e(a)b) = e(a) b, ale(a) = u, (alb)lb’= alb b’, and such that thejinite patchingproperty holds. (e is called the extent function.) The B-sheaves form a topos Sh(B, 8) over Z. The projection R x B + B is a B-sheaf (as R is regular) and is even a gfobaf B-sheaf as everyxcan be written asyje(x) wh’eree(y) = 1. R x Bisafield objectin Sh(B, 8) andis the generic field obtained from R.

Proof. This seems to be well known.

Remark. In view of the above, K(P) is the global sections object of a field F in Sh(B, Ssn) and this classifies field extensions for which P splits. In Sets, K(i) is a finite product of fields (from the proof of 3.1) and the Galois group is (locally) a subgroup of S,,. It remains to show that the information we want in general is expressible geometrically in terms of the Galois theater. Examples 6.1,6.2,6.4 illustrate what is involved.

Notation. In a commutative regular ring, Idem denotes the smallest idempotent for which x Idem =x.

Construction. Given K(i) and F as in the above remark we shall use the action of the root permutation group, S,,, to construct the Galois theater. We define d as the Galois theory and theaters of action in a topos 157

smallest relation such that the extent of d(f, jJ) exceeds b in B whenever:

(*) V Idem(axi - yi) Z ha(b) for all (T E S,.

Note that Vi Idem(cxi - yi) is a measure of the degree to which u distinguishes f from y. In Sets (as well as locally in many other cases) it suffices to compute d by patching minimal idemponents, b. Then either o(b) = b or bv(b) = 0, so d(Z, y) has “truth value” (or extent) at least b (for b minimal) whenever (J distinguishes .f from jj for the subgroup Gb of all u which fix b. In general, it is readily shown that this is the Galois theater of F over (the image of) K.

3.3. Proposition. Let K, F be as above and define d as indicated by (*). Define 0 by using the orbits of S,,. Then (d, 0) is a pre-theater and (when made into a theater) is Gal.Tht.(F/K).

Proof. That (d, 0) is a pre-theater in Sh(B, gsn) can be reduced to geometric conditions in 8 which hold as they are true if 8 is Sets. Similarly d “preserves the ring operations” and K is the fixed field. The result follows from 2.3.

4. The classifying topos

In this section we construct the classifying topos for theaters. Not unexpectedly this turns out to be Fun(T, Sets) where T is the category of finite theaters and the underlying set functor is the generic theater. Since the finite theaters are the same as the “finitely presented theaters” the result is a small extension of Coste’s work on classifying topoi, (21. (There are some pitfalls in generalizing this result to categories which “look like T,” see the remark following Theorem 4.7.) It follows that theaters in the Grothendieck topos 8 are the same thing as inverse image functors, N*, from Fun(T, Sets) to 8 which in turn are determined by their restrictions N = N*lToP. [Note that Top is Yoneda embedded in Fun(T, Sets).] Our first result characterizes those functors N: Top + 8 which extend to inverse image functors. If Top had finite limits then these would just be the left exact functors, but Top has few limits.

Remark. It is convenient to regard an object of T as a finite set A together with a group Adm(A) of admissible permutations of A as in 1.1. Then a T-morphism is a one-to-one functionf:A + B such that for each c E Adm(B) there existsf*(a) = T in Adm(A) for which f7 = af.

Definition. If A and B are in T, then C in T together with injections i:A + C, j: B + C is a T-union of A and B if C is the union of i(A) and j(B) and if Adm(C) is the groupof all permutationscr for which i*(cr) and j*(c) both exist and are admissible. 158 J.F. Kennison

If {Ck} is a representative family of T-unions, then it is a coproductfumify in the sense that the functor [A, -1 x [B, -1 is the disjoint union of [Ck, -1 in the obvious way.

Definition. Tr is the full subcategory of Fun(T, Sets) consisting of the mono- preseruingfunctors. We give TL the topology it inherits from the canonical topology on Fun(T, Sets). Then clearly, Sh(T”) = Fun( T, Sets). It is easily shown that T” is limit-closed in Fun(T, Sets). We let Tt be the finite limit-closure of Top in T”. We finally note that Sh(Tg ) = Fun(T, Sets) when Tr also inherits the canonical topology from Fun( T, Sets).

Definition. A functor N: Top-+ 8 is pre-feft exact iff N satisfies the following three conditions: (PLO) N preserves the terminal object of Top. That is N maps the empty theater to 1. (PLl) Let f and g be maps from A to B. If f and g are distinct then the equalizer of N(f) and N(g) is 0. (In other words, for 8 = Sets, b E N(B) and N(f)b = N(g)b imply f = g.) (PL2) If {C,} is a representative set of T-unions of A and B then N(A) x N(B) is the coproduct of {N(G)}.

4.1. Proposition. Let 8 be a Grothendieck topos. Let N: Top-, 8 be given. Then the following statements are equivalent: (i) N is pre-left exact; (ii) N has a continuous extension to T,” ; (iii) N has a continuous extension to T’; (iv) N extends to an inverse image functor N*: Fun( T, Sets) + %.) In view of 4.7, below, we shall be able to add: (v) N is of the form Hom(-, M) where M is a theater in 8 and Hom(-, M) is the object of theater homomorphism in 8 (each object A of T is finite so Hom(A, M) is determined by geometric conditions.)

Proof. The equivalence of (ii), (iii), (iv) follows from extension theorems such as [6, p. 371, and the above observations that Sh( Tr) = Sh( Tz ) = Fun( T, Sets). (iv)*(i). Assume (iv). Then N preserves any existing finite limits in Top so (PLO) follows. Next let f # g be maps from A to B. Then the zero functor is the equalizer in Fun(T, Sets) of f and g. (PLl) follows as N* must preserve the zero functor and equalizers. (PL2) follows since N* must preserve coproducts and finite products. (i)*(iii). Let N: Top-+ Sets be pre-left exact. (We have replaced 8 by the topos of Sets which is suflicient for our proof. For example, one could use Tf and apply Theorem 6.2.4 in [6, p, 1801). We let 15: Ty + Sets be the left Kan extension of N. Then fl preserves the canonical colimits so N is continuous iff fi is left exact. We need a definition and some lemmas to complete the proof. Galois theory and theaters of action in a topos 159

Definition. Let N: Top-, Sets be pre-left exact. Then (T, N) is the “comma cate- gory” whose objects are pairs (A, a) with A in T, a EN(A) and whose maps f: (A, a) --*(B, 0) are those f:A + B for which N(f)0 = (Y.Moreover, if X is in T” then (T, N, X) is the iterated comma category of 3-tuples (A, a, a) with A in T, a E N(A), aE X(A) and maps f: (A, a, a)+ (B, j3,b) being those f: A + B with N(f)/3 = a, X(f)a = 6.

4.2. Lemma. (i) Both (T, N) and (T, N, X) can be regarded us pre-ordered sets [with (A, (Y) c (B, /3) precisely when there is a necessarily unique morphism from (A, cu) to (B, /3) and similarly for (T, N, X)]. (ii) (T, N) has finite least upper bounds. (iii) I?(X) is the set of components of (T N, X) (regarded us a graph), where Nis the left Kan extension.

Proof. (i) Follows from (PLl). (ii) Let (A, a), (B, p) be given. Let C with i: A + C, j: B -, C be the unique T-union and y the unique element of N(C) for which N(i)y = cy and N(j)y = /3 (this uses (PL2)). Then (C, y) = (A, y) v (B, @). (iii) This is essentially the definition of the left Kan extension.

4.3. Lemma. Let (A, a, a), (B, p, b) be giuen in (T, N,X). Let (C, y)= (A, CY)v (B, fi) and i: A + C, j: B + Cbe us above. Then either (A, (Y, a) and (B, p, 6) huce no common upper bound orX(i)u =X(j)b in which case (C, y, c) = (A, CY,a) v (B, /3,b) where c is the common value ofX(i)a, X(j)b. Moreocer, (A, a, a) and (B, p, b) ure in the same component of (T, N, X) iff they huae a common upper bound.

Proof. If (D, 8, d) is a common upper bound of (A, (Y, a) and (B, p, 6) then there exists f: (C, y) + (D, 8) with X( fi)a = d = X( fj). The first part of the lemma follows as X(f) is mono since X E TX. The second part of the lemma follows mainly by showing that the property of having a common upper bound is a transitive relation. This reduces to a diagram chase using (PLl).

4.4. Corollary. (A, a, a,) is in the same component us (A, cr, a~) iff aI = az.

Proof. (A, (Y) = (A, cu) v (A, cu) and apply the above lemma.

4.6. Lemma. ti preserves equalizers.

Proof. In T” let the subfunctor W c X be the equalizer off, g: X + Y. Let (A, cy, a) represent an element of N(X). Let 6r = fA(a), b2 = gA(u). Then (A, o, a) is in the equalizer of 3(f), N(g) iff br = b2 (in view of 4.4 above) iff a E W(A) iff (A, ty, a) represents a member of I?(W). 160 J.F. Kennison

4.7. Lemma. N preserves finite products (including the terminal object which is an empty product). (This completes the proof of 4.1.)

Proof. The terminal object is preserved by (PLO). Given (A, a, a) in (T, N, X) and (B, p, b) in (T, N, Y) we associate (C, y, c) in (T, N, X Y) where (C, y) = (A, a) v (B, p) and c = (X(i)a, Y(j)b). Conversely given (C, y, c) we associate to it the pair (C, y, cl), (C, y, cz). It is easy to show that these associations produce the required equivalences between fl(X x Y) and N(X) x A( Y).

4.7. Theorem. Fun(T, Sets) classifies theaters, with the underlying set fun&or U: T + Sets being the generic theater. That is, theaters in Grothendieck topoi 8 correspond ro inverse image functors from Fun(T, Sets) to 25’.If the topos 8 has countable colimits, then the theaters in 8 correspond to preleft exact N: Top+ 2%‘.

Proof. In the usual way it suffices to prove our first assertion for %’= Sets and then observe that it generalizes. The rest of the theorem follows by examining the construction involved. If M is a theater in Sets we associate the pre-left exact functor N = Hom(-, M), where “Horn” refers to theater homomorphisms. We claim that N*(U) is iso- morphic to M. By the above, an element of N*(U) is represented by a 3-tuple (A, f, a) where A E T, a E A and f: A --,M. Lemma 4.4 says, in effect, that (A, f, a) is equivalent to (B, g, b) iff f(z) = g(b). Every element of M can be written as f(a) since the orbits are finite. Conversely, let N: Top-, Sets be pre-left exact and let h4 = N*(V). We claim that N(B) = Horn@, M). Given p E N(B) the map b + (B, p, 6) is in Horn@, M). Con- versely, let h: I3 -, M be given with h(b) represented by (A,,, ab, ab). Let (C, y) = V (Abr ab). Then h (6) can be represented by (C, y, Q,). Then b + cb is a homomor- phism ho: B + C. To h we associate p = N(ho)y.

Remark. Proposition 4.1 generalizes to any category C which resembles T. Under certain conditions, 4.7 also generalizes. We define a category C to be manic if: (i) Ail morphisms of C are mono. (ii) For X, Y in C there is a coproduct family {Zi} for which [X, -] x [ Y, -1 in the disjoint union of [Zi, -1. (iii) There is an inifial objecr family {Ai} such that the constant functor 1 on C is the disjoint union of [Ai, -1. For such categories pre-left exactness can be defined and the pre-left exact functors Cop + Sets correspond to inverse image functors Fun[C, Sets] + Sets. If C is count- able this extends to Grothendieck topoi. Theorem 4.7 does not always generalize. If the objects of C are models of a theory which are constructible in any reasonable topos and if each model of the theory is a union of models in C then the proof of 4.7 will (probably) go through. A precise generalization of 4.7 can be induced from the examples below. Galois theory and theaters of action in a topos 161

The category M of sets and monomorphisms and the subcategory MO of finite sets and monos are manic categories. MO classifies decidable objects in any topos. (If N: MO+ Zpis pre-left exact then N(1) x N(1) is the coproduct of N(1) and N(2) by (PL2). Then N(2) is the complement of the equality relation on N(l).) See also [4, Theorem 4.121, for a more general version. The category F of fields and the subcategory FO of those fields generated as fields by a finite set (of algebraic or transcendental elements) are both manic categories. FO classifies the models of a theory of fields enriched by unary relations of “having characteristic 0” and of “being transcendental” together with n-ary relations for each n of “being algebraically independent.” [One can also define the unary relation “having characteristic p” by the truth value of p = 0. The relations “having characteristic 0 [or p]” are consrunr relations or, in effect, 0-ary relations or truth values]. Over spatial topoi, FO classifies these sheaves of fields for which: (1) Every section has characteristic 0 on an open (hence clopen) set. (2) An n-tuple of sections is algebraically independent over an open (hence clopen) set.

5. Protinite groups

Let Go be the category of finite groups. We define a profinite group in 5%to be a left exact functor P: Go+ 2%The model homomarphisms (i.e., natural transformations) then go in the “wrong” direction, for profmite groups in Sets the model homomor- phisms coincide with the continuous homomorphisms except the domain and co-domain are interchanged. For this reason a profinite group in 8 is usually nor a group object (a profinite group in Fun[C, Sets] produces a group object in Fun[CoP, Sets]). The classifying topos Fun[T, Sets] for theaters has a canonical profinite group P: Go --, Fun[ T, Sets] defined by:

P(G)(A) = Group Hom[Adm A, G]

(where Adm A is the group of admissible permutations of A in the sense of the proof of 1.1. It is readily seen that Adm is a contravariant functor from T to Go.) It follows that every theater in any topos has an associated profinite group. Thus we can associate a profinite group to every Galoisian field extension. To get a Galois theory based on prolinite groups we need to find a way of identifying the “closed sub- groups.” In Sets, if Q is a closed subgroup of the profinite group P then Q defines a quotient functor of [P, -I: Go + Sets by equating f with g whenever f]Q = glQ. It is readily shown that Q is determined by this equivalence relation (and the fact that Q is closed), since points not in Q have open neighborhoods missing Q. (One must use the proof that epis in Go are onto.) It remains to find conditions telling us which quotients of [P, -1 arise from closed subgroups. We can require that the relation R = ((f, g>jfjQ = g/Q} be left exact as a functor from Go. In Sets this means that R is 162 J.F. Kennison

represented by a profinite group R. Thus we need to know conditions on R z P which make it the kernel pair of some P+ Q. (The arrows are dual to continuous homomorphisms.) We can, of course, require that R be an equivalence relation which leads to the following problem:

5.1. Problem. Are equivalence relations effective in the dual of the category of prolinite groups and continuous homomorphisms? Are there simple necessary and sufficient conditions for an equivalence relation to be effective?

We do note that there is a separable theory describing when R is effective. R would always be effective in the topos Fun[GO, Sets]. So, if H is the quotient P/R in Fun[GO, Sets] then it is necessary and sufficient that H be a subfunctor of some left exact functor (for then the reflection of H in the left exact functors (=Profinite groups) gives us the quotient in that category and if H is a subfunctor of its reflection one can show that R is effective). But, examining the argument in [5], H is such a subfunctor unless there is a finite configuration which forces two elements of H to be identified. There are only a countable number of basic configuration types so a countable number of coherent conditions would suffice.

6. Examples

Example 6.1. It is instructive to apply the process of Section 3 to the “problem” of splitting x2 + 1 over C, the complex numbers in Sets. Obviously we must get an isomorphic copy of C and Sets back again, but the isomorphism depends on a choice which is a less trivial choice in the related Examples 6.2, 6.4. C(r) is isomorphic to C x C which is a Zz-set (where the action transposes the components). The Boolean algebra of idempotents can be viewed as the algebra of clopens of the two point (with non-trivial &action). Therefore Sh(B, Setsz2) is the category of pairs (M, N) of sets together with an equivalence of M and N.

Example 6.2. Let K be a “twisted” sheaf of fields over the punctured plane (say the plane with the origin deleted). Let each stalk of K be the complex field with the solution sheaf of x2 + 1 = 0 being a non-trivial Zz-torsor. Assume that the real part of K is untwisted so that the ring of global sections is isomorphic to R, the reals. Then Sh(Z3, Szz) can be described as the topos of sheaves over X (where X is the two-to-one covering of the punctured plane) such that for each point p in the punctured plane there is an equivalence ep between Mp and ND the two stalks lying over p. It is required that if we glue Mp to Np via ep then we get a sheaf over the punctured plane. This is the expected equivalence with the original topos. The example is more interesting if we extend K to a sheaf over the entire plane by adding a real stalk at the origin. Then we clearly cannot add 4-1 at the origin (and stay in the Galois theory and [hearers of acrion in a roper 163 same topos) as we would need a local section at the origin satisfying x2 + 1 = 0. Because of the twisting there is no such section. Let Y be the of the (new) Boolean algebra B. Then Y is a connected space over the plane having one point over the origin and looking like X over the punctured plane. [Y is not a sheaf over the plane - this reflects the fact that Stone spaces cannot be internally constructed.] The new topos, in which x2+ 1 splits, might best be described as sheaves over Y together with the equivalence e, and a &-action on the stalk over the origin which is compatible with the ep action. This is a topos because there it represents a Wraith- Artin glueing of Zz-Sets onto the topos of sheaves over X (with ep). In general, the splitting of x2 + 1 can always be so represented. See 6.3 below.

Example 6.3 (J-1 adjuncrion, u sketch). Let K be a field in 8. For convenience assume that 1+ 1 is invertible. Let 0 be the open subtopos on which x2 + 1 splits. Let C be the closed subtopos. Let 8: 0 + C be the fringe functor. Represent K as (K,, K1, g: K1 + a(KJ). Let T be the solution torsor of x2 + 1 = 0 in Ko, Define a new topos by defining A: 0 + Cz2 b y A(X)=(3(Xr)) (with Z2-action induced by the action on T). Define a new field (Ko, K&l, G: K,[i]+ a(Kr)) where G(a +bi) = g(a)+g(b)T. CIn general the topos for a splitting field for this type of polynomial is obtained by glueing various pieces depending on how the polynomial factors.)

Example 6.4. Let P be an irreducible polynomial over a field K in Sets. Then K(i) is a product of fields Fr x Fz x * - * x F,. The action of S,, on K(i) forces all the factors Fi to be isomorphic as in 6.1. Then the relation d(M, N), as defined in Proposition 3.3 is readily examined by applying the equation (*) to the minimal idempotents bl, . . . , b, of K(i). At each i we need only consider those rr for which ub, = bi. This is the usual definition of the Galois group of Fi/K and (*) reduces to the expected condition that d(M, N) iff a(M) # N for all u in the Galois group. (The usual definition of a splitting field for P seems to be effectively equivalent to choosing an i and letting Fi be the splitting field.)

Example 6.5 (Theaters and profinite groups in spatial topoi). Let X be a topological space. We say that G is a relatively compactgroup overX if p: G +X is a group in the category of spaces over X and if G is “precisely as compact as X is” (meaning that if U is an ultrafilter on G and p*(U) the projected ultrafilter on X then for each limit x of p*(U) there is a unique limit of U lying over x. So if X is compact and Hausdorff, then G satisfies this condition iff G is compact and Hausdorff.) It follows that each fibre G, is compact and Hausdorff. We say that G is a profinite group over X if it is a relatively with profinite fibres. In general, G is not a sheaf over X but for each F we have that Hom(G, F) is a sheaf and “profinite group” has exactly the same meaning as given in Section 5 when applied to spatial topoi. 164 J.F. Kennison

Suppose that the relatively compact group G acts continuously and effectiveIy on the decidable sheaf over X. Then G is profinite and determines a theater structure on E. Every theater arises in this way. Therefore if K G F are sheaves of fields with F Galoisian over K, then the Galois groups of F over K is a profinite group and intermediate subfields correspond to closed subgroups.

Example 6.6. Let F be the constant sheaf, with every stalk being the complex field, over [0, 11.Let K c F be the subfield with KP = Fp for p # 1 and with Kr = the real field. Then F is Galoisian over K but the internal group of K-automorphisms of F is trivial. The profinite group, in the sense of 6.5 above, is not a sheaf, and has fibre Zz over 1 and trivial fibre elsewhere.

References

[t] J.C. Cole, The bicategory of topoi and spectra, J. Pure Appl. Algebra, to appear. [2] M. Coste, Localisation darts les categories de modeles, Theses, Universite Paris Nord, Paris, 1977. [3] P.T. Johnstone, Topos Theory, L.M.S. Mathematical Monographs no. 10 (Academic Press, New York, 1977). [4] P.T. Johnstone and G. Wraith, Algebraic theories in , in: Johnstone and Pare, eds, Indexed Categories and their Applications, Lecture Notes in Mathematics, Vol. 66 1 (Springer-Verlag, Berlin, 1978) 141-242. [5] J.F. Kennison, On limit-preserving functors, Illinois J. Math. 12 (1968) 616-619. [6] M. Makkai and G.E. Reyes, First Order Categorical Logic, Lecture Notes in Mathematics, Vol. 611 (Springer-Verlag, Berlin, 1977). [7] M. Tierney. Forcing topologies and classifying topoi, in: A. Heller and M. Tierney, eds, Algebra, Topology and , a collection of papers in honor of Samuel Eilenberg (Academic Press, New York, 1976) 211-219.